Bosonic code 

An error set relevant to bosonic stabilizer codes is the set of displacement operators, a bosonic analogue of the Pauli string basis for qubit codes. For a single mode, its elements are products of exponentials of the mode's position and momentum operators, acting on the mode's position states \(|y\rangle\) for \(y\in\mathbb{R}\) as \begin{align} e^{-iq\hat{p}}\left|y\right\rangle =\left|y+q\right\rangle \,\,\text{ and }\,\,e^{iq\hat{x}}\left|y\right\rangle =e^{iq y}\left|y\right\rangle ~, \tag*{(1)}\end{align} where \(q\in\mathbb{R}\). For multiple modes, error set elements are tensor products of elements of the single-oscillator error set, characterized by the vector of coefficients \(\xi\in\mathbb{R}^{2n}\).

The displacement error set is a unitary basis for trace-class linear operators on the \(n\)-mode Hilbert space that is Dirac-orthonormal under the Hilbert-Schmidt inner product [1]. There are two definitions of code distance associated with displacements. The definition inherited from qubit codes is the minimum weight of a displacement operator (i.e., number of nonzero entries in \(\xi\)) that implements a nontrivial logical operation in the code. The second definition is the minimum Euclidean distance (i.e., \(\ell^2\)-norm of \(\xi\)) such that the corresponding displacement implements a nontrivial logical operation in the code.

An error set relevant to Fock-state bosonic codes is the set of loss operators associated with the amplitude damping (a.k.a. photon loss or attenuation) noise channel, a common form of physical noise in bosonic systems. For a single mode, loss operators are proportional to powers of the mode's annihilation operator \(a=(\hat{x}+i\hat{p})/\sqrt{2}\), where \(\hat x\) (\(\hat p\)) is the mode's position (momentum) operator, and with the power signifying the number of particles lost during the error. For multiple modes, error set elements are tensor products of elements of the single-mode error set. A definition of distance associated with this error set is the minimum weight of a loss error that implements a nontrivial logical operation in the code.

An related error set is the set of powers of the Susskind–Glogower phase operator \(\frac{1}{\sqrt{a a^\dagger}} a\) and its adjoint [2–4] along with Fock-space rotations generated by the occupation number operator \(a^\dagger a\). These can also be obtained from qudit Pauli matrices through a limiting procedure [4] and allow one to expand trace-class operators despite not forming an orthonormal set [5]. These operators are correspong to the number-phase interpretation, a polar-like decomposition of a single mode, complementing the cartesian-like decomposition in terms of position and momentum displacements.